Construction and applications of Gaussian quadratures with nonclassical and exotic weight functions. (English) Zbl 1374.33016
Summary: In 1814 Carl Friedrich Gauss (1777–1855) developed his famous method of numerical integration which dramatically improves the earlier method of Isaac Newton (1643–1727) from 1676. Beside the some historical details in this survey, a formulation of this classical theory in modern terminology using theory of orthogonlity on real line, as well as the characterization, existence and uniqueness of these formulas, are presented. A special attention is devoted to the algorithms for constructing such quadrature formulas for nonclassical weight functions, their numerical stability and the corresponding software. Finally, some recent progress in this subject, as well as new important applications of these methods in several different directions (distributions in statistics and physics, summation of slowly convergent series, etc.) are presented.
MSC:
33C47 | Other special orthogonal polynomials and functions |
42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
41A55 | Approximate quadratures |
65D30 | Numerical integration |
65D32 | Numerical quadrature and cubature formulas |